Theorem nnssn0s 28472

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28466	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4092	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3985	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3904   ⊆ wss 3907  {csn 4585   0_s c0s 27956  ℕ_0scn0s 28463  ℕ_scnns 28464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-ss 3924  df-nns 28466

This theorem is referenced by:  nnssno  28473  nnn0s  28478  nnn0sd  28479  nnsgt0  28490